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G1=G x K1 has as vertices the simplices of G and a natural digraph structure. We show that dim(G1) is larger or equal than dim(G) and G1 is homotopic to G. The Kuenneth identity is proven using Hodge describing the harmonic forms by the product f g of harmonic forms of G and H and uses a d"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1505.07518","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-28T00:59:55Z","cross_cats_sorted":["cs.CG","cs.DM","math.AT"],"title_canon_sha256":"166f69357e44a909f0581b87b9da1442c3f421b9998a6fe2fdd301d68c180fb6","abstract_canon_sha256":"a47e6143b5a446503ced862246eaeb026f868da388b4c07784e24a0d03d063ca"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:00:58.160483Z","signature_b64":"C8viV8+YW3y59+12TD9wsao33UpGjNBUtrnQaOM78IXY03kIyc2ZV1G/chZdfSnkNOfIRzqDLPAYsgi2BPEyCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"789078df14144abd8ffaa6b0c96528d8d70d1648fc3ea59f36ca819ab04a9460","last_reissued_at":"2026-05-18T02:00:58.159797Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:00:58.159797Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Kuenneth formula for graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","cs.DM","math.AT"],"primary_cat":"math.CO","authors_text":"Oliver Knill","submitted_at":"2015-05-28T00:59:55Z","abstract_excerpt":"We construct a Cartesian product G x H for finite simple graphs. 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